Apolarity, Hessian and Macaulay Polynomials

A result by Macaulay states that an Artinian graded Gorenstein ring R of socle dimension one and socle degree δ can be realized as the apolar ring ℂ[∂/∂x0,...,∂/∂xn]/g⊥of a homogeneous polynomial g of degree δ in x0,..., xn. If R is the Jacobian ring of a smooth hypersurface f(x0,..., xn) = 0, then δ is equal to the degree of the Hessian polynomial of f. In this article we investigate the relationship between g and the Hessian polynomial of f, and we provide a complete description for n = 1 and deg(f) ≤4 and for n = 2 and deg(f) ≤3. © 2013 Copyright Taylor and Francis Group, LLC.